Complex Line Integrals

Consider the line integral of some function
of the complex variable taken (counter-clockwise) around a closed curve
in the complex
plane:

(6.158)

Because
, and writing
, where
and
are real functions,
it follows that
, where

(6.159)

(6.160)

However, we can also write the previous expressions in the two-dimensional vector form

(6.161)

(6.162)

where
,
,
,
, and
,
. According to
the curl theorem (see Section A.22),

(6.163)

(6.164)

where
is the region of the
-
plane enclosed by
. Hence, we obtain

(6.165)

(6.166)

Let

(6.167)

(6.168)

where
is a closed curve in the complex plane that completely surrounds the smaller curve
.
Consider

(6.169)

Writing
, a direct generalization of the previous analysis
reveals that

(6.170)

(6.171)

where
is now the region of the
-
plane lying between the curves
and
.
Suppose that
is well-behaved (i.e., finite, single-valued, and
differentiable) throughout
. It immediately follows that its real and imaginary components,
and
,
respectively, satisfy the Cauchy-Riemann relations, (6.17)-(6.18), throughout
.
However, if this is the case then it is apparent, from the previous two expressions, that
.
In other words, if
is well-behaved throughout
then
.

The circulation of the flow about some closed curve
in the
-
plane is defined

(6.172)

where
is the complex velocity potential of the flow, and use has been made of Equation (6.35). Thus, the circulation can be evaluated by performing a line integral in the complex
-plane. Moreover, as is clear from the previous discussion, this integral can be performed around any loop that can be continuously deformed into the loop
while
still remaining in the fluid, and not passing over a singularity of the complex velocity,
.